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Tuesday, October 21, 2008

Leading a Math Circle Proves at Least as Difficult as Expected

I finally had a first math circle meeting today. I invited any and all Math League students to come. Around 25 showed up at the beginning...8 or 10 had to leave in the middle and 2 more stopped in toward the end. I didn't know most of them but I think they were mostly juniors and seniors (in precalc and calculus). 6 or so students completely dominated the conversation.

I started with "what's the biggest number?" which quickly morphed into "are there different sizes of infinity?" and stalled at "is infinity a number?". They never agreed on whether or not infinity was a number, and by the end decided that they didn't even know what a number was. That was kind of neat. They also kept looking at me imploringly and saying "IS THERE AN ANSWER?!" and you all would have been proud of my aikido-like deflections.

For my part, I tried to steer them to focusing on the countable sets first, and broached the idea of one to one correspondence, but no one picked up that particular ball and ran with it.

So I guess I need help with -

Managing such a big group? More than 20 seems too big to guarantee everyone gets involved. I have 2 other teachers who are interested in "helping" but one of them could only come for about half the time and the other just stopped in said hi and left.

Encouraging dominant students to be more inclusive and collaborative?

Suggestions for steering them toward Cantorian set theory and the proofs I have in mind? I want to get them at least through the diagonal proof of the nondenumerability of the continuum, so that they can see that there is more than one distinct cardinality. At the end I postulated that there were as many positive even integers as there were positive integers but I didn't want to do too much. How should I start next week? I tried to pick a topic where I already knew "how it went", maybe that was a mistake, maybe I should have picked something unfamiliar.

It sounds awesome! And coincidentally, today in calculus we were talking about limits at infinity, and I got on a digression of infinity and the different "levels" of infinity.

I don't know *exactly* what a math circle is, but if this doesn't go against the math circle philosophy, maybe you could have students start out in smaller groups (putting one of the dominant speakers in each group) and have them individually grapple with the question. Maybe if they are not going in the direction you are hoping, you could also put some index cards on the table with some things for the groups to discuss: "how do you know there are the same number of even numbers as there are odd numbers?" or "is the number of whole numbers the same as the number of integers?" or "how do you know that the set containing {1, 5, 6} has the same number of elements as {2, 5, 10}? What about the set of all fifty states and the set of all fifty capitals."

But to keep the math circle really free and independent, only have groups that WANT to pick up a "discussion card" do it. Let the others go in whatever direction they want.

And then after a half hour or so, have the groups share what they discussed, and see if that sparks a more fruitful discussion?

Just an idea.

Also, this is a pretty good talk for a popular audience on infinity: http://www.youtube.com/watch?v=kBS_cNHvnBE

So you could also have them watch the first N minutes of the video, to get them to think, then before any "good" math starts, pause it and let them take it from there.

While trying to fall asleep last night I realized a glaring contradiction with what the kids were conjecturing that seems like a good starting point for next week. They were equally certain that:

1. infinity + 1 = infinity

and

2. the cardinality of the positive integers is smaller than the integers, because one is a subset of the other.

Which are of course mutually exclusive. Next week I'll be looking for ways to get them thinking about sizes of sets and applying one to one correspondence.

So thanks for the ideas - we will have to do something like smaller discussion groups if we get another large turnout. I like the idea of the prompt cards, and your prompting questions are excellent. The example I tried yesterday was "Are there more chairs in this room, or people? How do you know? Did you count them all?" - I think if a math circle has a philosophy it's something like "whatever works". :-) or maybe "give them enough rope to drive themselves into a contradiction".

There is merit in the idea that (for some appropriate definition of "smaller than") the set of positive integers is smaller than the set of integers. The problem, and the interesting thing, is the idea that cardinality and containment, while closely linked when you're talking about finite sets, become two different things when you're talking about infinite sets.

Informally I'll often refer to the "smallest" set that has some property, when a better word might be "minimal". In this context, I use the word "smallest" to mean "every other set with this property contains this one". One does have to be careful, though. I avoid the word "smallest" if there's any chance of ambiguity where it might be thought that every other set has a greater cardinality.

Hihi, if you get a chance, I'd love to hear your thoughts on the following as the year progresses with your math circle.

This year and last year I've been the adviser for math club and sometimes students come up with their own problems, and sometimes I give them problems. We only have a single 25 minute meeting each week, so we can't get *too* much done. Sometimes it takes us 2 or 3 weeks to finish a problem.

What I've noticed is that it's hard to keep the interest up on a problem after week 2. If they are on a good track, they think "oh we got it" and just decide to move on. If they are stuck (e.g. on the bloxorz problem), they just give up.

It's hard for us to delve deep into a topic.

I'm curious to see if you find the same thing, or if you have a way to avoid that?

These ideas came more from "Out of the Labyrinth" than direct experience, but this is my understanding about how they run the math circles in Cambridge:

The more you can get them explaining things to you, the better (keep resisting, so that they have to explain in detail). “I just don’t get it” is a very good line.

When you’ve all reached an impasse, say: “You know, we might just have reached the limits of human knowledge - perhaps this is just beyond our power to understand.” With the idea that they will take it as a challenge.